Anti-holomorphic Involution Research Articles

G2 mirrors from Calabi-Yau mirrors

We study the worldsheet CFTs of type II strings on compact G2 orbifolds obtained as quotients of a product of a Calabi-Yau threefold and a circle. For such models, we argue that the Calabi-Yau mirror map implies a mirror map for the associated G2 varieties by examining how anti-holomorphic involutions behave under Calabi-Yau mirror symmetry. The mirror geometries identified by the worldsheet CFT are consistent with earlier proposals for twisted connected sum G2 manifolds.

Real Slices of ${\rm SL}(r,\mathbb{C})$-Opers

Through the action of an anti-holomorphic involution $\sigma$ (a real structure) on a Riemann surface $X$, we consider the induced actions on ${\rm SL}(r,\mathbb{C})$-opers and study the real slices fixed by such actions. By constructing this involution for different descriptions of the space of ${\rm SL}(r,\mathbb{C})$-opers, we are able to give a natural parametrization of the fixed point locus via differentials on the Riemann surface, which in turn allows us to study their geometric properties.

Twistor geometry of the Flag manifold

A study is made of algebraic curves and surfaces in the flag manifold $$\mathbb {F}=SU(3)/T^2$$ , and their configuration relative to the twistor projection $$\pi $$ from $$\mathbb {F}$$ to the complex projective plane $$\mathbb {P}^{2}$$ , defined with the help of an anti-holomorphic involution $$j$$ . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space $$\mathbb {P}^3$$ of the 4-dimensional sphere $$S^4$$ . Deformations of twistor fibers project to real surfaces in $$\mathbb {P}^{2}$$ , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in $$\mathbb {F}$$ of bidegree $$(1,1)$$ . These surfaces define orthogonal complex structures on specified dense open subsets of $$\mathbb {P}^{2}$$ relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree $$(1,1)$$ are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in $$\mathbb {F}$$ .

Invariant hypercomplex structures and algebraic curves

AbstractWe show that ‐invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in correspond to algebraic curves C of genus , equipped with a flat projection of degree k, and an antiholomorphic involution covering the antipodal map on .

2d mathcal{N} = (0, 1) gauge theories and Spin(7) orientifolds

We initiate the geometric engineering of 2d mathcal{N} = (0, 1) gauge theories on D1-branes probing singularities. To do so, we introduce a new class of backgrounds obtained as quotients of Calabi-Yau 4-folds by a combination of an anti-holomorphic involution leading to a Spin(7) cone and worldsheet parity. We refer to such constructions as Spin(7) orientifolds. Spin(7) orientifolds explicitly realize the perspective on 2d mathcal{N} = (0, 1) theories as real slices of mathcal{N} = (0, 2) ones. Remarkably, this projection is geometrically realized as Joyce’s construction of Spin(7) manifolds via quotients of Calabi-Yau 4-folds by anti-holomorphic involutions. We illustrate this construction in numerous examples with both orbifold and non-orbifold parent singularities, discuss the role of the choice of vector structure in the orientifold quotient, and study partial resolutions.

Branes and moduli spaces of Higgs bundles on smooth projective varieties

Given a smooth complex projective variety M and a smooth closed curve \(X\, \subset \, M\) such that the homomorphism of fundamental groups \(\pi _1(X)\, \longrightarrow \, \pi _1(M)\) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context.

HYPERBOLIC GROUPS AND NON-COMPACT REAL ALGEBRAIC CURVES

In this paper we study the spaces of non-compact real algebraic curves, i.e. pairs (P, τ), where P is a compact Riemann surface with a finite number of holes and punctures and τ: P → P is an anti-holomorphic involution. We describe the uniformisation of non-compact real algebraic curves by Fuchsian groups. We construct the spaces of non-compact real algebraic curves and describe their connected components. We prove that any connected component is homeomorphic to a quotient of a finite-dimensional real vector space by a discrete group and determine the dimensions of these vector spaces.

Einstein metrics, conformal curvature, and anti-holomorphic involutions

Building on previous results [17, 35], we complete the classification of compact oriented Einstein 4-manifolds with \(\det (W^+) > 0\). There are, up to diffeomorphism, exactly 15 manifolds that carry such metrics, and, on each of these manifolds, such metrics sweep out exactly one connected component of the corresponding Einstein moduli space.

Mironov Lagrangian cycles in algebraic varieties

We generalize a construction due to Mironov. Some time ago he presented new examples of minimal and Hamiltonian minimal Lagrangian submanifolds in and . His construction is based on the considerations of a noncomplete toric action of , where , on subspaces that are invariant with respect to the action of a natural antiholomorphic involution. This situation takes place for a rather broad class of algebraic varieties: complex quadrics, Grassmannians, flag varieties and so on, which makes it possible to construct many new examples of Lagrangian submanifolds in these algebraic varieties. Bibliography: 4 titles.

Moduli spaces of vector bundles with fixed determinant over a real curve

Let $$(\Sigma ,\tau )$$ denote a Riemann surface of genus $$g \ge 2$$ equipped with an anti-holomorphic involution $$\tau $$. In this paper we study the topology of the moduli space $$M(r,\xi )^\tau $$ of stable Real vector bundles over $$(\Sigma ,\tau )$$ of rank r and fixed determinant $$\xi $$ of degree coprime to r. We prove that $$M(r,\xi )^{\tau }$$ is an orientable and monotone Lagrangian submanifold of the complex moduli space $$M(r,\xi )$$ so it determines an object in the appropriate Fukaya category. We derive recursive formulas for the $${\mathbb {Z}}_2$$-Betti numbers of $$M(r,\xi )^\tau $$ and compute $${\mathbb {Z}}_p$$-Betti numbers for odd p through a range of degrees. We deduce that if r is even and $$ g>>0$$, then $$M(r,\xi )^{\tau }$$ and $$M(r,\xi ')^{\tau }$$ have non-isomorphic cohomology groups unless $$\xi $$ and $$\xi '$$ have equivalent Stieffel–Whitney classes modulo automorphisms of $$(\Sigma ,\tau )$$. If r is even, and $$g>>0$$ is even, we prove that the Betti numbers of $$M(r,\xi )^{\tau }$$ distinguish topological types of $$(\Sigma , \tau ; \xi )$$. If $$r=2$$ and g is odd, we compute all $${\mathbb {Z}}_p$$-Betti numbers of $$M(2,\xi )^\tau $$.

Spin structures and branch divisors on $p$-gonal Riemann surfaces

We study spin structures on Riemann and Klein surfaces in terms of divisors. In particular, we take a closer look at spin structures on hyperelliptic and $p$-gonal surfaces defined by divisors supported on their branch points. Moreover, we study invariant spin divisors under automorphisms and antiholomorphic involutions of Riemann surfaces and count them. We generalize a formula that gives $2$-spin divisors, proved by Mumford, to the case of $m$-spin divisors for an even $m$, supported on branch points of a hyperelliptic surface.

Gluing construction of compact $\operatorname{Spin}(7)$-manifolds

We give a differential-geometric construction of compact manifolds with holonomy $\operatorname{Spin}(7)$ which is based on Joyce's second construction of compact $\operatorname{Spin}(7)$-manifolds and Kovalev's gluing construction of compact $G_2$-manifolds. We provide several examples of compact $\operatorname{Spin}(7)$-manifolds, at least one of which is new. Here in this paper we need orbifold admissible pairs $(\overline{X}, D)$ consisting of a compact Kähler orbifold $\overline{X}$ with isolated singular points modelled on $\mathbb{C}^4/\mathbb{Z}_4$, and a smooth anticanonical divisor $D$ on $\overline{X}$. Also, we need a compatible antiholomorphic involution $\sigma$ on $\overline{X}$ which fixes the singular points on $\overline{X}$ and acts freely on the anticanoncial divisor $D$. If two orbifold admissible pairs $(\overline{X}_1, D_1)$, $(\overline{X}_2, D_2)$ and compatible antiholomorphic involutions $\sigma_i$ on $\overline{X}_i$ for $i=1,2$ satisfy the gluing condition, we can glue $(\overline{X}_1 \setminus D_1)/\langle\sigma_1\rangle$ and $(\overline{X}_2 \setminus D_2)/\langle\sigma_2\rangle$ together to obtain a compact Riemannian 8-manifold $(M, g)$ whose holonomy group $\operatorname{Hol}(g)$ is contained in $\operatorname{Spin}(7)$. Furthermore, if the $\widehat{A}$-genus of $M$ equals 1, then $M$ is a compact $\operatorname{Spin}(7)$-manifold, i.e. a compact Riemannian manifold with holonomy $\operatorname{Spin}(7)$.

Character varieties for real forms

Let $\Gamma$ be a finitely generated group and $G$ a real form of $\mathrm{SL}_n(\mathbb{C})$. We propose a definition for the $G$-character variety of $\Gamma$ as a subset of the $\mathrm{SL}_n(\mathbb{C})$-character variety of $\Gamma$. We consider two anti-holomorphic involutions of the $\mathrm{SL}_n(\mathbb{C})$ character variety and show that an irreducible representation with character fixed by one of them is conjugate to a representation taking values in a real form of $\mathrm{SL}_n(\mathbb{C})$. We study in detail an example: the $\mathrm{SL}_n(\mathbb{C})$, $\mathrm{SU}(2,1)$ and $\mathrm{SU}(3)$ character varieties of the free product $\mathbb{Z}/3\mathbb{Z} * \mathbb{Z}/3\mathbb{Z}$.

Higgs Bundles, Branes and Langlands Duality

Given a compact connected Riemann surface X equipped with an anti-holomorphic involution and a complex semisimple Lie group G equipped with a real structure, we define anti-holomorphic involutions on the moduli space of G-Higgs bundles over X. We describe how the various components of the fixed point locus match up, as one passes from G to its Langlands dual LG. As an example, the case of $${G\,=\, {\rm SL}(2,{\mathbb C})}$$ and $${^LG\,=\,{\rm PGL}(2,\mathbb{C})}$$ is investigated in detail.

Real Holomorphic Sections of the Deligne–Hitchin Twistor Space

We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered.

Two-Dimensional Periodic Schrödinger Operators Integrable at an Energy Eigenlevel

The main goal of the first part of the paper is to show that the Fermi curve of a two-dimensional periodic Schrodinger operator with nonnegative potential whose points parameterize the Bloch solutions of the Schrodinger equation at the zero energy level is a smooth M-curve. Moreover, it is shown that the poles of the Bloch solutions are located on the fixed ovals of an antiholomorphic involution so that each but one oval contains precisely one pole. The topological type is stable until, at some value of the deformation parameter, the zero level becomes an eigenlevel for the Schrodinger operator on the space of (anti)periodic functions. The second part of the paper is devoted to the construction of such operators with the help of a generalization of the Novikov-Veselov construction.

Conformal automorphisms of algebraic surfaces and algebraic curves in the complex projective space

We study the automorphism group of curves and surfaces in CP3 with respect to the conformal group, i.e. the group of all A∈PGL(4,ℂ) commuting with the anti-holomorphic involution j defined by j((z0:z1:z2:z3))=(−z¯1:z¯0:z¯3:−z¯2). For some singular surfaces we check when this group is finite. Among the singular surfaces we handle there are:(1) certain cones;(2) surfaces X containing no line and with j(X)≠X;(3) surfaces containing only finitely many, k, twistor lines with k≥3.In many cases the proofs need results on conformal automorphisms of singular curves.

Algebraic families of groups and commuting involutions

Let [Formula: see text] be a complex affine algebraic group, and let [Formula: see text] and [Formula: see text] be commuting anti-holomorphic involutions of [Formula: see text]. We construct an algebraic family of algebraic groups over the complex projective line and a real structure on the family that interpolates between the real forms [Formula: see text] and [Formula: see text].

Symmetric products of a real curve and the moduli space of Higgs bundles

Abstract Consider a Riemann surface X of genus g ≥ 2 equipped with an antiholomorphic involution τ . This induces a natural involution on the moduli space M ( r , d ) of semistable Higgs bundles of rank r and degree d . If D is a divisor such that τ ( D ) = D , this restricts to an involution on the moduli space M ( r , D ) of those Higgs bundles with fixed determinant O ( D ) and trace-free Higgs field. The fixed point sets of these involutions M ( r , d ) τ and M ( r , D ) τ are ( A , A , B ) -branes introduced by Baraglia and Schaposnik (2016). In this paper, we derive formulas for the mod 2 Betti numbers of M ( r , d ) τ and M ( r , D ) τ when r = 2 and d is odd. In the course of this calculation, we also compute the mod 2 cohomology ring of S y m m ( X ) τ , the fixed point set of the involution induced by τ on symmetric products of the Riemann surface.

On the connectedness of the set of Riemann surfaces with real moduli

The moduli space $${\mathcal {M}}_{g}$$ , of genus $$g\ge 2$$ closed Riemann surfaces, is a complex orbifold of dimension $$3(g-1)$$ which carries a natural real structure, i.e. it admits an anti-holomorphic involution $$\sigma $$ . The involution $$\sigma $$ maps each point corresponding to a Riemann surface S to its complex conjugate $$\overline{S}$$ . The fixed point set of $$\sigma $$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $$\mathrm {Fix}(\sigma )$$ is the locus $${\mathcal {M}}_{g}(\mathbb {R})$$ , the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppala, and R. Silhol. The complement $$\mathrm {Fix}(\sigma )-{\mathcal {M}}_{g}(\mathbb {R})$$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $$\mathrm {Fix}(\sigma )$$ is connected.